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Normally spells that, for example, affect a 30' burst affect a sphere centered around a point chosen within range. What if you pick a point near the edge of your range, such that the appropriately shaped sphere would extend beyond the edges? Does the spell efffectively have a larger maximum range in such cases?

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No

Actually, emanation spells cast near the edge of their range take on a weird shape that is the set of spaces they would have affected less those outside the spell's range. The edge of such a region can be found by tracing the line of interpenetration between the spell's ordinary region of effect and a sphere whose radius is the range of the spell centered at the caster.

This is a consequence of the rules for spells' range descriptors:

Range

A spell’s range indicates how far from you it can reach, as defined in the range entry of the spell description. a spell’s range is the maximum distance from you that the spell’s effect can occur, as well as the maximum distance at which you can designate the spell’s point of origin. If any portion of the spell’s area would extend beyond this range, that area is wasted.

(emphasis added)

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    \$\begingroup\$ Wow, I must admit, I had never encountered that bit of text before. Not what I understood. \$\endgroup\$
    – YogoZuno
    Jan 15, 2018 at 23:21
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    \$\begingroup\$ That's definitely part of the rules that are houseruled in every game I run. \$\endgroup\$ Jan 16, 2018 at 15:16

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